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Contract No. DE-AC36-08GO28308

Technical Manual for the SAM Physical Trough Model Michael J. Wagner and Paul Gilman

Technical Report NREL/TP-5500-51825 June 2011

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Contract No. DE-AC36-08GO28308

Technical Manual for the SAM Physical Trough Model Michael J. Wagner and Paul Gilman Prepared under Task No. SS10.1110

Technical Report NREL/TP-5500-51825 June 2011

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PrefaceThis technical manual was developed by the authors to document the engineering principlesand methodology underlying the Physical Trough model in the System Advisor Model(Version 2011.5.4). The Physical Trough model was designed to provide performance analysiscapability for the parabolic trough concentrating solar power technology using systemgeometry and physical properties as input. This allows the modeler to assess performancewithout (always) requiring pre-existing empirical formulations for the system that may not beavailable for all systems of interest.

This model is intended to supplement existing trough modeling tools such as the SystemAdvisor Empirical Trough model that - as the name implies - approaches system performancemodeling from a “top-down” empirical perspective. Users may find different modelingapproaches useful throughout the various stages of the feasibility analysis and projectdevelopment process. An empirical modeling approach works best for systems withwell-known performance curves, or when detailed subsystem models are available and can becorrelated and rolled into simplified relationships.

Alternatively, the first-principles or “physical” modeling approach can be better suited forsystems where performance is unknown a priori. Accordingly, the authors emphasize that bothmodeling approaches are equally viable, although empirical and physical models answerfundamentally different questions and should be used within their inherent constraints. Theauthors also note that this document only presents one possible model formulation forparabolic troughs; many alternative and equally valid modeling solutions exist, and theapproach selected here does not necessarily represent the best possible one.

A complex engineering system such as the parabolic trough technology requirescorrespondingly complex performance modeling tools, and with complexity comes anopportunity for misunderstanding or misuse on the part of the modeler. The authors havedeveloped this technical manual to provide model transparency for the SAM user and todocument the motivation and justification for the various performance equations used in thePhysical Trough model. This manual presents equations, formulations, coding logic, andsystem descriptions as they appear in the performance model, although some “translation” isrequired between computer code and textual discussion. All major subsystems are discussed indetail, including the solar field, power cycle, thermal storage, fossil backup, and controlinterface.

This document is intended as a companion to the System Advisor help material, and theauthors recommend that new users become familiar with the Help material in addition toreviewing this document. The content is catered to technical users with an engineering orphysics background who are interested in understanding the System Advisor Physical Troughmodel on a more fundamental level. Non-technical users or those new to concentrating solarmay find this manual instructive in becoming familiar with the engineering principles behindthese systems.

iv

AcknowledgmentsThe authors acknowledge the invaluable contribution of the System Advisor team for theirfeedback and discussion throughout the model development and documentation process.Special thanks go to Nate Blair (NREL) for technical guidance, and to Aron Dobos and SteveJanzou (NREL) for their software integration support. The document benefited significantlyfrom the thorough review and input of Mark Mehos (NREL), Ty Neises (NREL), and CraigTurchi (NREL).

The authors thank Chuck Kutscher (NREL), Frank Burkholder (NREL), Greg Kolb (Sandia),Cliff Ho (Sandia), David Kearney (Kearney & Associates), Mike Erbes (Enginomix), HankPrice (Abengoa Solar), Diego Arias (Abengoa Solar), and Bruce Kelly (Abengoa Solar), all ofwhose insight, guidance, and previous work were valuable in the model formulation process.Finally, the authors acknowledge Tobias Hirsch (DLR), Markus Eck (DLR), Jürgen Dersch(DLR), Jan Fabian Feldhoff (DLR), Daniel Benitez (Flabeg), and the SolarPACES Task I teamfor enlightening discussion on parabolic trough systems and modeling methodologies.

Mike WagnerNational Renewable Energy Laboratory

v

Table of Contents

1 Introduction 11.1 TRNSYS simulation framework ................................................................ 11.2 Variable Names and Abbreviations............................................................. 11.3 Background and Modeling Approach.......................................................... 6

2 Solar Field 72.1 Energy balance ...................................................................................... 7

2.1.1 Nodal Energy Balance .................................................................. 82.2 Solar Field Control ................................................................................. 12

2.2.1 Collector defocusing..................................................................... 142.2.2 Field freeze protection .................................................................. 162.2.3 Accounting for transient effects....................................................... 16

2.3 Collector Assembly and Field Optics .......................................................... 192.3.1 Determining an average efficiency value ........................................... 26

2.4 Receivers (HCE’s) .................................................................................. 272.4.1 Modeling approach....................................................................... 282.4.2 Model formulation ....................................................................... 31

2.5 Piping model ......................................................................................... 392.5.1 Piping diameter selection ............................................................... 402.5.2 Piping layout and components ........................................................ 462.5.3 Performance calculations ............................................................... 48

3 The Power Cycle 523.1 General description................................................................................. 52

3.1.1 Modeling Approach...................................................................... 533.1.2 “Basis” Rankine cycle................................................................... 54

3.2 Regression model ................................................................................... 553.2.1 Regression modeling concepts ........................................................ 563.2.2 Off-design response framework....................................................... 573.2.3 Parameter normalization ................................................................ 573.2.4 Experimental design ..................................................................... 583.2.5 Formulating a model..................................................................... 583.2.6 Other calculated output ................................................................. 61

3.3 Heat rejection ........................................................................................ 633.3.1 Performance considerations............................................................ 643.3.2 Condenser performance limitations .................................................. 663.3.3 Wet cooling model ....................................................................... 683.3.4 Dry cooling model ....................................................................... 723.3.5 Parallel hybrid cooling model ......................................................... 73

3.4 Power cycle implementation ..................................................................... 763.4.1 Normal operation ......................................................................... 763.4.2 Shutdown and cold startup ............................................................. 773.4.3 Standby operation ........................................................................ 78

vi

3.4.4 Warm restart conditions................................................................. 79

4 Thermal storage and plant control 804.1 Thermal storage ..................................................................................... 80

4.1.1 TES Sizing ................................................................................. 814.1.2 Model formulation ....................................................................... 824.1.3 Storage freeze protection ............................................................... 85

4.2 Indirect storage heat exchanger ................................................................. 864.2.1 Heat exchanger sizing ................................................................... 874.2.2 Heat exchanger performance model ................................................. 88

4.3 Auxiliary heater ..................................................................................... 904.4 Plant control.......................................................................................... 91

4.4.1 Controller background and approach ................................................ 914.4.2 Available mass flow...................................................................... 924.4.3 Converting to thermal energy.......................................................... 934.4.4 Control logic............................................................................... 954.4.5 Operating modes.......................................................................... 974.4.6 Controller iteration .......................................................................101

5 Parasitic Losses 1055.1 Losses modeled in System Advisor ............................................................1055.2 Parasitic loss calculations.........................................................................106

5.2.1 SCA drives .................................................................................1065.2.2 Power cycle HTF pumps................................................................1065.2.3 Storage HTF pump.......................................................................1065.2.4 Fixed parasitic losses ....................................................................1075.2.5 Balance of plant parasitics .............................................................1075.2.6 Auxiliary heater operation..............................................................107

5.3 Practical considerations ...........................................................................1085.4 Net production at design ..........................................................................108

A Glossary of Terms 112

vii

List of Tables

1 Abbreviations used in this document .......................................................... 22 Variable naming conventions .................................................................... 33 Variable subscript conventions .................................................................. 54 General definitions for each fixed optical loss term ........................................ 265 Inputs to the receiver model ...................................................................... 306 Values of the mean free path between collisions of a molecule for free molecular

convection ............................................................................................ 327 Selection of coefficients C and m for Zhukauskas’ correlation based on the

Reynolds number at D3............................................................................ 338 Bracket geometry and material properties assumed by System Advisor for con-

ductive heat loss calculations .................................................................... 379 Pipe sizing schedules used in the trough model. The wall thickness and schedule

are selected to match a pressure rating of 25 bar. ........................................... 4510 The assumed piping lengths for the steam generator and pumping system. .......... 4611 Variable assignments for the piping equipment by domain. Referenced equa-

tions are denoted with square brackets. The pressure drop coefficients for thevarious piping components are defined in the kΔP column................................ 47

12 Design-point conditions for the basis Rankine cycle. The conditions were se-lected to match common parabolic trough operating conditions. ....................... 55

13 The experimental design for characterizing power block performance. The ex-periment is adapted from a three-level full-factorial design. ............................. 59

14 Summary of inputs to the wet cooling model. ............................................... 6815 Property functions used by the wet cooling model. ........................................ 6916 Summary of inputs to the dry cooling model. ............................................... 7217 Double-check criteria for the converged power cycle solution under normal op-

eration. ................................................................................................ 7718 A summary of the parasitic losses accounted for by System Advisor .................105

List of Figures

1 One possible field arrangement, where the field is broken up into two headersections. Each loop in this illustration contains 8 individual SCA’s, and eachportion of the header is connected to two loops - one on the top and one on thebottom of the image. ............................................................................... 7

2 The nodal structure of the loop is shown (left) where each SCA in the loop isan autonomous node. This framework allows multiple receiver/collector types -shown as A and B (center) - and user-specified defocusing schemes (right). ........ 8

3 Energy balance for the receivers in an SCA. The enclosed box represents thecontrol volume, encompassing the HTF within the absorber tubes and pipingfor a single SCA..................................................................................... 8

viii

4 Hypothetical situation where neither the final temperature, nor the average ofinitial and final temperatures gives an accurate representation of the temperatureover the time step. This situation can be encountered in dynamic systems likeCSP plants. ........................................................................................... 17

5 The trough includes both a collector to reflect light and a receiver to absorb andtransport heat......................................................................................... 19

6 The angle between the solar irradiation and the normal vector to the collectoraperture plane. ....................................................................................... 20

7 Focal length geometry for calculating the average focal length. ........................ 238 Two adjacent collector rows may shadow each other if the tracking angle is

sufficiently severe. The shadowing is subject to the collector aperture width, therow spacing (centerline to centerline), and the tracking angle of the collectors. .... 24

9 A heat balance for the modeled receiver. Heat transfer in the radial direction(left to right) is considered, while circumferential and axial transfer is not. ......... 28

10 The thermal resistance network for the “intact” receiver model shown in Figure9. Energy is absorbed at T3 and T4−5........................................................... 29

11 The thermal resistance network for the “broken glass” receiver model. Energyis absorbed on the absorber tube surface at T3 and heat is exchanged directlywith the sky and ambient temperatures. ....................................................... 29

12 An illustration of the flow diverting from the header into individual field loops.In this example, two loops extract HTF per section. The diameter of the headeris adjusted to maintain optimal flow velocities. ............................................. 41

13 Three possible field configurations modeled by System Advisor. Two (top left),four (bottom left), and six (right) field subsections are shown. .......................... 42

14 Numbering scheme and notable mass flow rates for the subfield and runner piping. 4415 The solution for Eq.[2.108] at X = 0 is shown for three unique Re values. .......... 4916 Control volume near the power cycle condenser. The power cycle control vol-

ume bisects the condenser to include the steam flow but not the cooling flow. ...... 5317 Schematic for simplified basis Rankine cycle ............................................... 5418 Data showing how the power and heat output varies with condenser pressure

[bar] at three distinct levels of HTF inlet temperature. .................................... 6019 Data showing how the power and heat output varies with HTF temperature at

three distinct levels of HTF mass flow rate (top), and the associated interactioneffects (bottom)...................................................................................... 61

20 A simplified Rankine cycle diagram. Heat is added to the cycle by the solarfield via a heat exchanger. Heat is removed from the system via a wet coolingtower. Power output is equal to the heat addition minus the heat rejection. .......... 64

21 Dry and wet bulb temperatures for five summer days in an arid climate. Peaktemperature differences approach 20◦C during the hottest hours of the day.......... 65

22 The impact of a 20◦C temperature change on condenser pressure and cycle per-formance. ............................................................................................. 66

23 Steam velocity as a function of steam pressure through ductwork in a plantdesigned for a minimum condenser pressure of 0.068 bar ............................... 67

ix

24 An illustration of the various temperature rises that influence the condenser pres-sure for a wet-cooled system. These include the ambient wet-bulb temperature,the approach temperature, the cooling water temperature rise, and the heat ex-changer hot side temperature difference....................................................... 70

25 An illustration of the various temperature rises that influence the condenser pres-sure for a dry-cooled system. These include the ambient dry-bulb temperature,the air temperature rise, and the heat exchanger hot side temperature difference. .. 73

26 An illustration of the parallel wet/dry cooling configuration is shown. The ACC(left) and wet system (right) share the heat rejection load. ............................... 74

27 The trough power plant subsystems include the solar field, piping, thermal stor-age, auxiliary heating, and the power cycle. ................................................. 80

28 A schematic of the variable-volume tank model. Fluid level varies with differ-ences in the inlet and outlet mass flow rate. The outlet temperature is equal tothe average of the fully mixed tank. ............................................................ 82

29 Conceptual representation of the tank control volume. .................................... 8330 The indirect storage heat exchanger is defined by the hot and cold side approach

temperatures. A sketch of the physical system arrangement (left) correspondsto a plot of the temperatures in the heat exchanger as a function of position (right). 86

31 The control flow diagram for the physical trough model. .................................10332 The net and gross power over a period of four days. .......................................109

x

1 IntroductionThis manual defines and documents the conventions, methodology, and information flowassociated with the System Advisor Model (SAM) physical trough model in the SystemAdvisor Model [10]. A detailed overview of each subsystem is provided, with theengineering/mathematical basis laid out where appropriate. While this documentation containsgeneral information on the model structure, the beginning modeler may find portions of thecontent challenging. For an introductory overview on how to use the model, please refer to theSystem Advisor help system. Definitions of technical terms (denoted with italic text) areprovided in the Glossary of Terms at the end of this document (page 112).

The System Advisor Model provides a consistent framework for analyzing and comparingpower system costs and performance across the range of solar technologies and markets, fromphotovoltaic (PV) systems for residential and commercial markets to concentrating solarpower and large PV systems for utility markets.

System Advisor is based on an hourly simulation engine that interacts with performance, cost,and finance models to calculate energy output, energy costs, and cash flows. The software canalso account for the effect of incentives on cash flows. System Advisor’s spreadsheet interfaceallows for exchanging data with external models developed in Microsoft® Excel. Most ofSystem Advisor’s inputs can be used as parametric variables for sensitivity studies toinvestigate impacts of variations in performance, cost, and financial parameters on modelresults.

1.1 TRNSYS simulation frameworkSystem Advisor models system performance using the TRNSYS1 software developed at theUniversity of Wisconsin combined with customized components. TRNSYS is a validated,time-series simulation program that can simulate the performance of photovoltaic,concentrating solar power, water heating systems, and other renewable energy systems usinghourly resource data. TRNSYS is integrated into System Advisor so there is no need to installTRNSYS software or be familiar with its use to run Solar Advisor.

The default source code folder \SAM\\exelib\trnsys\source containsthe FORTRAN code for each TRNSYS module: sam_mw_trough_Type250.f90 (solar field),sam_mw_trough_Type251.f90 (storage and dispatch), sam_mw_pt_Type224.f90 (powercycle), and the shared HTF property subroutines sam_mw_pt_Type229.f90 andsam_mw_pt_propmod.

1.2 Variable Names and AbbreviationsThe following tables contain information and conventions used in this document. Table 1 liststhe abbreviations, Table 2 lists variable naming conventions, and Table 3 list commonly usedvariable subscript conventions.

1For more information on the TRNSYS software, refer to documentation from the University ofWisconsin at sel.me.wisc.edu/trnsys/default.htm

1

Table 1: Abbreviations used in this document

Abbreviation DescriptionACC Air-cooled condenserCSP Concentrating Solar PowerDNI Direct-normal irradiationHTF Heat transfer fluidIAM Incidence angle modifierIOCop Inlet/outlet/cross-over pipeITD Initial temperature differenceLHV Lower heating valueNREL National Renewable Energy LaboratorySAM System Advisor ModelTES Thermal energy storageTOU Time of use

2

Table 2: Variable naming conventions

Name Description Units Units Abbrv.A Area meters squared m2c, cp Specific heat Joules per kilogram Kelvin JkgKC Calculation coefficient varies variesĊ Thermal capacity rate Watts per Kelvin WKD Diameter meters mE Energy Joules Jg Gravitation constant meters per sq. second ms2f Fraction none nonef r Friction factor none noneh Enthalpy Joules per kilogram JkgH Height meters mi, j Counting indexes none noneIbn Direct solar irradiation Watts per sq. meter Wm2k Thermal conductivity Watts per meter-Kelvin Wm·KL Length meters mm Mass kilograms kgṁ Mass flow rate kilograms per second kgs(mc) Thermal inertia Joules per Kelvin JKN Integer number none noneNu Nusselt number none noneP Pressure Pascals PaPr Prandtl number none noneq̇ Heat transfer rate Watts WQ̇ Cycle heat absorption rate Watts Wr Ratio none noneR Radius meters mR̂ Thermal resistance Watts per Kelvin WKRa Rayleigh number none noneRe Reynolds number none nones Entropy Joules per kilogram Kelvin JkgT Temperature Kelvin KT Average temperature Kelvin Kt Time seconds su Specific internal energy Joules per kilogram JkgU Internal energy Joules JUA Thermal conductance Watts per Kelvin WKv Velocity meters per second msV Volume meters cubed m3Ẇ Cycle power output Watts W

Continued...

3

Continued...

Name Description Units Units Abbrv.α Absorptance none noneβ Volumetric expansion coef. inverse Kelvin 1KΔ Change in value none noneγ Thermal loss coefficient Watts per sq. meter Kelvin Wm2·Kγsol Solar azimuth degrees ◦δ Solar declination degrees ◦η Efficiency none noneθ Aperture incidence angle degrees ◦θe/θz Solar elevation/zenith degrees ◦ε Emittance / Effectiveness none noneμ Dynamic viscosity Pascal-seconds Pa · sν Kinematic viscosity sq. meters per second m

2

sπ Pi none noneρ Density kilograms per cubic meter kgm3τ Transmittance none noneφ Latitude degrees ◦χ Thermodynamic quality none noneψ Longitude degrees ◦ω Hour angle degrees ◦

4

Table 3: Variable subscript conventions

0 Previous value / initial value1..9 Item n in a sequence1tank Single tankA Availableabs Absorbedad j Adjustedair Ambient airamb Ambientap Apertureapproach Approach temperatureaux Auxiliary (fossil)ave Averagebal Balance-of-plantbd Blowdownboil Steam boilerbrac Bracketc Coldcap Capacitycalc Calculatedchg Chargingcol Collectorcond Conduction / condenserconv Convectioncs Cross-sectionalcw Cooling watercycle Power cycledb Dry-bulbde f Defocusdem Demanddes Design pointdis Dischargingdri f t Condenser driftdump Dumped energyduty Heat exchanger dutye Electricenv Envelopef Focalf an Cooling fanf in Finalf sec Field sectionsf uel Fossil fuelg Guessgross Gross electricf p Freeze protectionh Hothdr Header

hdrsec Header sectionhgrp Header section groupshl Heat losshsec Header sectionsht f Heat transfer fluidhyd Hydraulichx Heat exchangerin Inletinc Incidentloop All SCA’s in a loopLHV Lower heating valuem Mirrormin Minimummax MaximumND Non-dimensional, normalizednet Net electricopt Opticalout Outletp Pumppar Parasiticpb Power block (or power cycle)pm Per meter basisrad Radiationrec Receiverre j Rejectedrun Runner piperunsec Runner sections Isentropicsby Standbysca Solar collector assemblyset point Design point or setpoints f Solar fieldsol Solarspacing Row spacingst Steamstart Startupsys Systemtes Thermal energy storageth Thermaltot Totaltou Time of usetrack Collector trackingtrans Transientw Waterwb Wet-bulb↑ / ↓ Increase / Decrease

5

1.3 Background and Modeling ApproachThe physical trough model characterizes a parabolic trough CSP plant by derivingperformance equations from first principles of heat transfer and thermodynamics wherepossible. In practice, this means that empirical “curve-fit” relationships are eliminated to thedegree that is practical for the type of modeling analysis done in System Advisor. The primarybenefit of this approach is the added flexibility in changing system parameters and componentproperties at a fundamental level (i.e. absorber emissivity, glass thickness, etc.) and simulatingtheir impact on overall system performance. With this increased flexibility come a fewdrawbacks; these include the addition of multiple layers of modeling uncertainty and theincreased opportunity for divergence in the results from a real system. While an empiricalmodel can produce high accuracy over the range of parameters used in its development, suchmodels cannot provide predictive performance outside these ranges. Both the physical andempirical trough models are included in System Advisor to allow for comparisons between thephysical and empirical modeling approaches.

Besides fulfilling the goal of deriving system performance from first principles, the physicalmodeling approach achieves several other objectives: the model includes transient effectsrelated to the thermal capacity of the HTF in the field piping, headers, and the balance of theplant; it allows for more flexible field component specification, including multiple receiver andcollector types within a single loop; it maintains a reasonably short run-time allowing forparametric and statistical analyses; and it makes use of existing models where possible.Previously existing subsystem models that are adapted and incorporated into the physicalmodel include the collector model from System Advisor’s empirical trough model, the receiverheat loss model [8], a field piping pressure drop model [14], and the power cycle performancemodel [20] originally developed for System Advisor’s power tower CSP system model. Thesemodels are discussed in more detail in dedicated sections of this report.

6

2 Solar FieldThe solar field is the heat-collecting portion of the plant. It consists of one or more loops ofsolar collector assemblies (SCA’s), with each loop laid out in parallel. A common header pipeprovides each loop with an equal flow rate of heat transfer fluid (HTF), and a second headercollects the hot HTF to return it either directly to the power cycle for power generation or tothe thermal energy storage system for use at a later time. To minimize pumping pressurelosses, the field is typically divided into multiple sections, each section with its own header set,and the power cycle is situated near the middle of the field. Figure 1 shows one possible plantlayout where two header sections are used for 20 total loops.

Figure 1: One possible field arrangement, where the field is broken up into two header sections.Each loop in this illustration contains 8 individual SCA’s, and each portion of the header is

connected to two loops - one on the top and one on the bottom of the image.

2.1 Energy balanceWithin each loop, a number of SCA’s are used to incrementally heat the HTF to the designoutlet temperature. Each SCA is composed of a number of parabolic collectors and theirreceivers2 in series that share a single common tracking drive. In this model, the SCA servesas the lowest level of discretization. Each SCA is treated as an independent calculation nodewithin the loop, and the absorbed energy, losses, temperature, pressure drop, and otherperformance values are calculated independently for each SCA. This allows each SCA toimpact performance separately and potentially allows each SCA to contain different receiverand/or collector attributes. System Advisor allows the user to specify unique geometry andperformance characteristics for each SCA in the loop up to a limit of four uniqueconfigurations. The order in which the SCA’s are defocused in the loop during high-fluxconditions can also be modified under this framework. Figure 2 illustrates these principles.

2The term “receiver” in this model is interchangeable with “heat collection element” (or HCE) thathas been used in other models

7

Figure 2: The nodal structure of the loop is shown (left) where each SCA in the loop is anautonomous node. This framework allows multiple receiver/collector types - shown as A and B

(center) - and user-specified defocusing schemes (right).

2.1.1 Nodal Energy BalanceA typical steady-state receiver model determines the temperature rise across the node byconsidering the absorbed energy, the mass flow rate of HTF through the receiver, and thespecific heat of the HTF. This energy balance for node i is represented in Eq.[2.1].

ΔTi = Tout,i−Tin,i =q̇abs

ṁht f cht f(2.1)

However, in the case of the parabolic trough technology, the thermal inertia associated with theenergy state of the node can impact performance to the extent that a steady-state model isinsufficient; thus transient terms must be included. The most significant transient effect in thesolar field is the thermal mass of the HTF in the headers and in the receiver piping, so we needto consider the change in energy of the HTF in deriving the energy balance equations. We’llstart with the energy balance for a single SCA node drawn in Figure 3.

Figure 3: Energy balance for the receivers in an SCA. The enclosed box represents the controlvolume, encompassing the HTF within the absorber tubes and piping for a single SCA.

8

The energy balance above shows a receiver tube with an inlet flow, outlet flow, absorbedenergy (net absorption is positive, net loss is negative), and an internal energy term. The q̇ heatflows are functions of the mass flow rate (ṁht f ) which is constant across the boundary, and thetemperatures Tin and Tout . The average nodal temperature T is equal to the average of the inletand outlet temperatures of the calculation node, since the temperature rise across the node isassumed to be linear.

The internal energy term ∂U∂t represents the change in energy of the node as a function of timet. Equivalently, this term can be expressed as:

∂U∂t

=(m cht f +(mc)bal,sca L

) ∂T∂t

(2.2)

Here, m is the mass of the HTF contained in the node, L is the length of a single SCA, and cht fis the specific heat of the HTF. An additional thermal inertia term (mc)bal,sca is included toaccount for the thermal mass of piping, joints, insulation, and other SCA components thatthermally cycle with the HTF. The (mc)bal,sca L term is dimensionally equivalent to the HTFcapacitance term m · cht f , even though it is input into the model as a single value. Thisconvention eliminates the need for specific knowledge of either the exact mass or specific heatof the additional thermal inertia while still allowing the user to account for the inertia effect.This term is dimensionally defined to represent the amount of thermal energy per meter ofcollector length required to raise the temperature of the node one degree K. The input units areWt−h/m−K, and the value is converted within the code to units of J/m−K.

Note that in terms of the mathematical formulation, the HTF and material properties areassumed to be constant. However, when evaluating system performance, material and HTFthermal properties are evaluated as functions of temperature (and pressure, if applicable).

The total energy balance in the control volume is:

q̇in+ q̇abs =∂U∂t

+ q̇out (2.3)

The inlet and outlet heat flows can be expressed as:

q̇in− q̇out = ṁht f cht f (Tin−Tout)= 2 ṁht f cht f

(Tin−T

)(2.4)

Substituting this equation and the definition for the internal energy term into the energybalance equation, then solving for the first differential, the result is:

∂T∂t

=2 ṁht f (Tin−T )+ q̇absm cht f +(mc)bal,sca

(2.5)

9

This is a linear first order differential equation, which has the general solution:

T =q̇abs

2 ṁht f · cht f+C1 exp

[−

2 ṁht f cht fm cht f +(mc)bal,sca

Δt]+Tin (2.6)

This equation has an unknown constant C1 that can be determined by enforcing a boundarycondition. In this situation, we know that the average nodal temperature T = T 0 at thebeginning of the time step when t = 0, and we define T 0 to be the temperature T at the end ofthe previous time step (the average temperature is T = (Tout+Tin)/2). Thus, solving for theunknown constant C1:

T 0 = T |t=0

=q̇abs

2 · ṁht f cht f+C1e0+Tin

C1 = T 0−q̇abs

2 · ṁht f cht f−Tin (2.7)

Finally, we substitute the constant into the general solution to find the final equation for theoutlet temperature from each SCA shown in Eq.[2.8].

For i= 1,Nsca:

Tout,i = 2 T i−Tin,i

Tout,i =q̇abs,i

ṁht f cht f ,i+Tin,i

+2 ·(T 0,i−

q̇abs,i2 · ṁht f cht f ,i

−Tin,i)exp[

−2 ṁht f cht f ,i Δtmi cht f ,i+mci,bal,sca Li

](2.8)

This equation is applied to each node i in the loop, where Tin,i is equal to the outlet temperatureof the previous node in the loop, Tout,i−1. Since the calculated temperature for each nodedepends on both the inlet temperature of the previous node and the node temperature from theprevious time step, these values must be established as boundary conditions. The temperatureof the node at the previous time step is stored from time step to time step, and the inlettemperature is set equal to the outlet temperature of the previous node for each but the firstnode in the loop to satisfy these requirements. The HTF mass of each node is calculated as afunction of the receiver piping volume and the local HTF density.

For i= 1,Nsca:mi = ρht f ,i Li Acs,i (2.9)

The inlet temperature at the first node, representing the inlet of the entire field, requiresadditional consideration. In a similar derivation process as the one described in Figure 3, a“system” temperature is calculated for both the hot and cold sides of the solar field. The cold

10

system temperature is used as the node #1 inlet temperature, and the hot system temperature isused as the effective solar field outlet temperature. These values incorporate the thermal inertiaassociated with the header and balance-of-plant HTF mass. Under steady-state conditions, theloop inlet HTF temperature equals either:

• the power cycle outlet temperature

• the storage loop outlet temperature

• a mass-flow weighted average of the storage and power cycle outlet temperatures

• or the solar field outlet temperature (during nighttime recirculation)

depending on the control situation. However, directly using any of these outlet temperatures asthe loop inlet value is inaccurate because it fails to account for the thermal inertia of theheader. If we include thermal inertia as a transient effect, the derived equation for loop inlettemperature (denoted Tsys,c) is shown in Eq.[2.10].

Tsys,c = (Tsys,c,0−Ts f ,in) exp

⎡⎣− ṁht fV c ·ρc+

(mc)bal,ccc

Δt

⎤⎦+Ts f ,in (2.10)

Here, the cold header temperature from the last time step is Tsys,c,0, the volume in the coldheader and the runner pipe is given by Vc, and cold fluid density is ρc. Analogously, the hotsystem outlet temperature combines loop outlet flow, the header and runner pipe volumes, andthe state of the system at the last time step.

Tsys,h = (Tsys,h,0−T ∗loop,out) exp

⎡⎣− ṁht fV h ·ρh+

(mc)bal,hch

Δt

⎤⎦+T ∗loop,out (2.11)

Here, T ∗loop,out represents the heat-loss adjusted loop outlet temperature calculated in Eq.[2.26]. The capacity term (mc)bal,h (or (mc)bal,c for the cold header) is used to account for anythermal inertia that isn’t included in the HTF volume calculations. Varying this termeffectively adds or subtracts internal energy capacity to the HTF in the system; its best use is asan empirical adjustment factor to help match observed plant performance. This variable isdefined in terms of the thermal power (kilowatt hours) per gross electricity capacity(Megawatts) needed to raise the header temperature one degree Celsius. The (mc)bal,h term isapplied specifically to the temperature calculation for the hot header, while (mc)bal,c describesonly the additional cold header thermal inertia.

To illustrate this concept, consider the following example. A hypothetical 100MWe (gross)trough plant with a 35% conversion efficiency is observed starting up from a cold overnighttemperature of 140◦C. The system temperature increases a total of 150◦C to 290◦C after 1hour of running the solar field at full load. A modeler predicts from known HTF inventory thatthe plant should have reached its startup temperature in only 30 minutes. Thus, the (mc)bal

11

http:Eq.[2.10

terms must together account for an half of the full-load hour that the plant took to start up.Assuming both coefficients are equal, the hot-side coefficient would be calculated as follows:

(mc)bal,h =0.52 ·0.35

·100,000[kWe]

100[MWe] ·150[K](2.12)

(mc)bal,h = 4.8[

kWhrMWecap ·K

]

Realistically, the value calculated here will be lower once the thermal inertia of the solar fieldpiping, insulation, and joints are considered. This calculation is only provided as ademonstration of the procedure for determining an empirical thermal inertia term, and the usershould carefully select their input to model their desired system.

Section 2.1 Summary

• The model calculates solar field temperatures and corresponding performance values inde-pendently at each SCA specified in the loop.

• Each SCA can have a different associated collector or receiver, up to four in total.

• The performance of one loop is representative of each loop in the field.

• The model determines the solar field inlet temperature by accounting for the return flowfrom storage/power cycle/field outlet and the volume of the cold header and piping.

• The model determines the solar field outlet temperature by accounting for the hot flow fromthe collection elements, the volume of the hot header and piping, and it adds user-specifiedthermal capacity terms that impact the transient behavior of the system.

2.2 Solar Field ControlCSP plants convert direct normal irradiation (DNI) from the sun into thermal energy andeventually into electricity or some other useful product. DNI can vary significantly over arelatively short period of time, and the solar field in a trough plant must be designed to handlethese variations. The solar field control algorithm uses user-specified input to make operationaldecisions based on the DNI resource level, ambient temperature, presence of thermal storage,etc. The minimum HTF mass flow rate, the maximum HTF mass flow rate, the HTF outlettemperature, defocus state, and the temperature of each node relative to the freeze-protectionset point are parameters that are monitored and enforced by the field controller.

The field logic employs an “ideal” control strategy. This means that the HTF mass flow rate iscontrolled to allow the loop outlet temperature to meet the design point value when possible.Receiver heat loss and surface temperature are both functions of the HTF temperature, and

12

HTF temperature in turn is a function of HTF mass flow rate, so the HTF temperature must becalculated iteratively. System Advisor uses successive substitution, where an initial estimatefor the field mass flow rate is provided along with guessed temperature values, then the massflow rate is recalculated and adjusted until the outlet temperature converges to the design value.

If the mass flow rate is calculated to fall outside of the acceptable range specified by the user,special control handling is required. The minimum and maximum allowable loop flow ratesare specified indirectly by the user on the System Advisor Solar Field page through anallowable HTF velocity range. Eq.[2.13] shows how the minimum (vht f ,min) and the maximum(vht f ,max) HTF velocities are converted to mass flow rates.

ṁht f ,min = vht f ,min ρht f ,c π(Dmin2

)2

ṁht f ,max = vht f ,max ρht f ,h π(Dmin2

)2(2.13)

Here, ρht f represents the HTF density at the cold (c) and hot (h) design temperatures, andDmin is the minimum diameter receiver tube that is present in the system.

The solar field mass flow rate is constantly adjusted within the upper and lower flow limits tomaintain the desired HTF outlet temperature. Several methods of calculation are possible fordetermining the mass flow rate. System Advisor uses a customized numerical solutionalgorithm that considers the rate of change in the mass flow rate and HTF outlet temperaturevariables from iteration to iteration. Numerical techniques generally require an initial guessvalue, and this is the case for System Advisor. The initial mass flow guess is shown in Eq.2.14.

ṁht f ,guess =∑Nscai=1 q̇i,abs Nloops

cht f ,ave (Tloop,out,des−Ts f ,in)(2.14)

Note that the average specific heat value refers to the integral average over the length of thecollector loop, the HTF outlet temperature is the solar field design outlet temperature, and theinlet temperature is equal to the value provided by the power block or fluid source. Subsequentiterations depart from this energy balance basis by using the mass flow and temperature valuesfrom the previous iteration (ṁ′ht f and T

′loop,out , respectively), and the values from the current

iteration (ṁht f and Tloop,out) to predict the mass flow for the next iteration (ṁ∗ht f ). Thiscalculation is shown in Eq.[2.15].

ṁ∗ht f = ṁht f +(ṁht f − ṁ′ht f ) ·

(Tloop,out,des−Tloop,outTloop,out −T ′loop,out

)(2.15)

13

http:Eq.[2.15http:Eq.[2.13

where:

−0.75 |ṁht f | ≤

[(ṁht f − ṁ′ht f ) ·

(Tloop,out,des−Tloop,outTloop,out −T ′loop,out

)]≤ 0.75 |ṁht f | (2.16)

The limits of [−.75,+.75] in Eq. [2.16] were selected to promote model convergence and donot reflect any physical limit on the bounds of the mass flow rate. The limits apply fromiteration-to-iteration within a single time step, while time-step-to-time-step variations in themass flow rate may exceed these limits.

2.2.1 Collector defocusingIf the calculated mass flow rate through the loop falls below the minimum allowable flow rate(ṁht f ,min), then the code resets the flow rate to be equal to this minimum value and recalculatesthe outlet temperature of the loop, assuming a fixed flow rate. Thus, in conditions where theminimum criterion is not met, the outlet temperature from the field will fall below the designvalue.

In the opposite case, the solar field may absorb so much energy that the mass flow raterequired to maintain the design outlet temperature exceeds the maximum specified value. Oncethe mass flow rate reaches an upper limit, the only way to avoid over-temperature HTF (and adangerous situation) is to reduce the amount of absorbed energy by defocusing collectors. Theplant controller can defocus collectors either when the amount of energy produced by the solarfield exceeds what can be consumed in the power cycle and/or storage, or when the convergedmass flow rate exceeds the value calculated in Eq.[2.13]. For the latter case, the logic considersthe total amount of absorbed energy in each SCA in the field and defocuses the SCA’s until thetotal absorbed energy falls below the threshold required by the maximum mass flow rate.

SCA’s are defocused using one of three user-selected schemes.

Full SCA DefocusingThe first option allows the user to specify an order in which each SCA in a loop should bedefocused. During defocusing periods, the SCA’s will be completely defocused in successiveorder until the total absorbed power falls below the imposed limit. Note that thermal lossesincurred by the defocused SCA(s) are still accounted for in all energy calculations. Eq.[2.17]shows how the code determines the number of collectors to defocus. The total effectivethermal output power qs f ,eff is reduced by the amount absorbed in each collector according todefocus order until the maximum allowable thermal output qs f ,limit is met.The array items A[i]indicate the order of defocusing requested by the user, and the subscripts abs, hl, and inc referto the absorbed thermal energy, thermal heat losses, and irradiation on each SCA, respectively.

For i= 1,Nsca:And while q̇s f ,e f f > q̇s f ,limit and i

Then, for j = 1, i:q̇i,inc = 0

This option is activated by clearing the Allow partial defocusing check box on the Solar Fieldpage.

Partial Sequenced DefocusingThe second option also allows sequenced defocusing of SCA’s, but instead of completelydefocusing the SCA’s, partial defocusing is allowed. Sometimes called feathering, this practiceis implemented in real plants by modulating the tracking angle to partially shift the receiver outof the reflected radiation, thus reducing the intercept factor. This control scheme sequentiallydefocuses the SCA’s according to the order specified on the Solar Field page, but allows oneSCA to be defocused partially. The defocusing calculations in this mode require three steps:first, the number of affected SCA’s is determined, second, the SCA’s that are wholly unneededare completely defocused, and third, the final defocused SCA is partially modulated to matchthe desired thermal output.

For i= 1,Nsca:And while q̇s f ,e f f > q̇s f ,limit and i

Note that the defocusing and control algorithms are iterative; consequently, the variablesappearing on both sides of the equations from Eq.[2.17] through Eq.[2.20] are provided withinitial values and are iteratively modified until the solution converges.

The model tracks and reports the total defocused energy (also called “dumped” energy). Thisvalue measures the amount of incident energy that is not allowed to reach the collector as aresult of defocusing, and the reported value includes dumped energy caused by both the fieldcontroller and the general plant controller. Dumped energy is approximated by considering theproduct of the total aperture area (Aap,tot) collector optical efficiency during the time step(ηopt), the solar irradiation (Ibn) and the fraction of defocused SCA’s during the time step(ηde f ). Thermal losses are not included in the dumped energy calculation.

q̇dump = Aap,tot Ibn ηopt ηde f (2.21)

2.2.2 Field freeze protectionDuring times of extended shutdown or cool nighttime temperatures, the heat transfer fluid inthe solar field may cool to an unacceptably low temperature. To avoid this situation, electricheat trace equipment can provide supplemental heat to the HTF in the solar field. SystemAdvisor models this situation by enforcing the minimum field HTF temperature specified asthe freeze protection temperature on the Solar Field page. The temperatures in each node of theloop and in the headers are monitored. If the temperature falls below the minimum allowablevalue, heat is added to the system to maintain the temperature at the minimum value. Thisenergy is tracked and reported as a parasitic loss. The magnitude of freeze protection energy iscalculated for each of the i nodes in the loop of Nsca collector assemblies in Eq.[2.22].

Ef p =Nsca∑i=1

((Tf p−Tht f ,ave,i) Acs,i Li ρht f cht f + q̇i,hl

)·Nloops (2.22)

The total freeze protection energy Ef p can be divided by the time step to determine the rate ofenergy consumption. In Eq.[2.22] the average node temperature is represented by Tht f ,ave,i, thereceiver tube cross-sectional area is Acs,i, the length of the receiver tube for the node is Li, anddensity and specific heat properties of the HTF are used. Likewise, the freeze protectionenergy required for the headers and runner piping is calculated in the following general form.

Ef p,hdr =(Tf p−Thdr

)· (Vhdr+Vrun) ρ cp+ q̇hl,hdr+ q̇hl,run (2.23)

Electric heat tracing is assumed for the freeze protection parasitic.

2.2.3 Accounting for transient effectsUnlike traditional utility-scale power plants, CSP systems are subject to frequent andsignificant temporal fluctuations in the thermal resource. Fossil or nuclear plants will spendmost of their lifetime operating near design conditions, but CSP plants vary significantly inoutput over relatively small time periods. Thus, the impact of transient effects may become

16

http:Eq.[2.22http:Eq.[2.22http:Eq.[2.20http:Eq.[2.17

significant in the overall performance of the plant. Simply including transient terms in theformulation of the system model (see Eq.[2.8] for example) may not be sufficient to adequatelyaccount for their impact. Instead, averaged equations that account for plant behavior over theduration of the time step are required.

Numerical simulations rely on an approximation that assumes that a continuous variablesurface can be discretized into finite homogeneous sections. This is the case for the calculationof loop temperature depicted in Figure 2 where the collector loop is discretized according tothe number of SCA’s in the loop. Likewise, the time variable t that is assumed to be continuousin the model formulation must be discretized for the numerical simulations performed bySystem Advisor. The challenge for CSP simulations that rely on hourly weather information isthat the discretized time step is often much longer than what is required to change the state ofthe CSP system. The state of the system at the end of the 1-hour time step may not reallyresemble conditions through most of the hour. The 1-hour time step is long enough thatcalculating the solar field mass flow based on the final system temperature, or even using theaverage temperature, would lead to significant inaccuracies. Figure 4 illustrates this difficulty.

Figure 4: Hypothetical situation where neither the final temperature, nor the average of initial andfinal temperatures gives an accurate representation of the temperature over the time step. This

situation can be encountered in dynamic systems like CSP plants.

To remedy this situation, System Advisor adjusts the total absorbed energy by subtracting theamount of energy that contributed to changing the energy state of the plant (i.e. the “transientenergy”). The transient energy is calculated in Eq.[2.24].

Etrans =(Vhdr,h ρht f ,h cht f ,h+(mc)bal,h

)(Tsys,h−Tsys,h,0)

+(Vhdr,c ρht f ,c cht f ,c+(mc)bal,c

)(Tsys,c−Tsys,c,0) (2.24)

+Nsca∑i=1

(Acs,i Li ρht f ,i cht f ,i+Li (mc)bal,sca,i

)· (T̄i− T̄i,0) ·Nloops

Equation [2.24] consists of three parts; the first and second terms measure the energy thatcontributes to changing the “hot” and “cold” parts of the system, respectively (i.e. the header

17

http:Eq.[2.24

HTF volume and piping between the solar field and the power cycle), while the third termmeasures the energy contributing to changing the temperature of the HTF, piping, andinsulation in the collector loops.

Included in the thermal inertia calculations are the hot and cold header volume plussupplemental runner pipe (Vhdr) and the user-specified thermal mass terms (mc)bal,h and(mc)bal,c described previously. The collector thermal inertia term represents the sum of HTFmass for all collectors in all loops, and is increased by the length-specific thermal inertia term(mc)bal,sca,i also discussed previously. The energy state is relative to the temperature from theprevious time step - (Tsys,h,0) for the hot header and (Tsys,c,0) for the cold header.

The energy value calculated in Eq.[2.24] is subtracted from the total absorbed energy in thesolar field to determine the total available energy3 from the solar field during the given timestep. The mass flow rate is recalculated based on the energy available from the field.4

ṁA,s f =q̇A,s f

cht f ,ave (Tsys,h−Ts f ,in)(2.25)

Where:q̇A,s f = q̇s f −

EtransΔt

The temperature Ts f ,in represents the inlet HTF temperature to the solar field and Tsys,h is thereturn temperature to the power cycle calculated in Eq.[2.11]. The return temperaturecalculation presented in Eq. [2.11] uses the heat-loss adjusted loop outlet temperature ratherthan the temperature at the immediate outlet of the field loops. Thermal losses from the headerand runner piping are accounted for as part of the loop outlet temperature by using theuser-specified piping loss coefficient λhl from the Parasitics page and the total transport pipingsurface area Apipe,tot calculated by the model.

T ∗loop,out = Tloop,out −λhl Apipe,tot

(Tloop,out −Tamb

)ṁht f cht f ,ave

(2.26)

3Note that the term “available” is not used in a thermodynamic sense, but simply to indicate that aportion of the absorbed energy from the field is not available for use in the power cycle or storage becauseof internal energy changes within the field.

4In practical terms, the mass flow and thermal energy values calculated in Eq.’s [2.25, 2.26] representthe effective achieved values over the time step. These values are reported to the power block, controlalgorithm, and thermal storage system for subsequent calculation. These are also the values that arereported in the System Advisor hourly performance results.

18

http:Eq.[2.11http:Eq.[2.24

Section 2.2 Summary

• The solar field controller limits the mass flow rate of HTF through each loop by defocusingcollectors in over-temperature conditions or by overriding the outlet temperature require-ment in low-resource conditions.

• Collector defocusing is achieved by either completely defocusing the SCA’s in the sequencespecified by the user, or by partially defocusing the SCA’s in sequence or simultaneously.

• Freeze protection is applied according to the minimum allowable temperature specified bythe user.

• Transient effects are applied by recalculating the mass flow rate from the field according toabsorbed energy minus energy that goes into changing the energy state of the HTF.

• Piping thermal losses are determined based on the hot outlet temperature from the solarfield.

2.3 Collector Assembly and Field OpticsThe collector model and optical calculations used in the physical trough model are based onthe collector model in System Advisor’s empirical trough model. System Advisor defines thecollector as the portion of the solar field that reflects irradiation to the receiver. This equipmentis distinct from the receiver component that consists of an evacuated glass envelope and tubereceiver, as shown in Figure 5. The optical calculations for the collector model extend to thepoint of determining the magnitude of solar flux that is incident on the receiver.

Figure 5: The trough includes both a collector to reflect light and a receiver to absorb andtransport heat.

To determine the flux incident on the receiver, we must consider both constant optical lossesand variable optical losses that change with solar position. The total irradiation on the field is afunction of the equivalent aperture area of all of the collectors in the field, the strength of thesolar insolation, and the angle at which the irradiation strikes the aperture plane. Theequivalent aperture area refers to the total reflective area of the collectors that is projected onthe plane of the collector aperture. This area is distinct from the curved reflective surface. Areathat is lost due to gaps between mirror modules or non-reflective structural components is notincluded in the aperture area value. Thus, though the structure of the collector may occupy100m lengthwise and 5m across the aperture, for example, the total reflective aperture area

19

may be somewhat less than 100×5= 500m2 after spaces, gaps, and structural area areaccounted for.

When the solar irradiation is not normal to the plane of the collector aperture, losses areincurred that scale with the acuteness of incidence angle. The incidence angle θ is equal to thedifference in angles between the normal to the aperture plane and the solar irradiation. This isdepicted in Figure 6.

Figure 6: The angle between the solar irradiation and the normal vector to the collector apertureplane.

The incidence angle is a function of solar position and the collector tracking angle. To find thesolar position for a particular plant at latitude φ and longitude ψ, we first calculate the solartime tsol [5].

B = (day−1)360365

EOT = 9.2 (0.000075+0.001868 cosB−0.032077 sinB−0.014615 cos(2B)−0.04089 sin(2B))

shi f t = zone ·15−ψ

tsol = hour+shi f t15

+EOT60

(2.27)

EOT is an equation of time that determines the deviation in local time from solar time as afunction of the day of the year, shi f t represents the fixed time-shift in minutes due to thedifference between the standard longitude for the time zone (zone, negative zones to the westand positive to the east of GMT+0) and the longitude at the location of interest. The day of theyear (day) and the hour of the day (hour, from 0 to 24) are also required. The time of day isconverted into an hour angle (ω) in Eq.[2.28].

ω = (tsol−12) ·15◦ (2.28)

Since the solar position is impacted by the tilt of Earth’s axis, the declination angle δ must also

20

http:Eq.[2.28

be determined.δ = 23.45◦ · sin

(360

(284+day)365

)(2.29)

Finally, the solar azimuth (γsol) and the solar elevation (θe) angles are calculated [5].

θe = sin−1 (sin(δ) sin(φ)+ cos(φ) cos(δ) cos(ω)) (2.30)θz = 90◦ −θe

γsol = sign(ω)∣∣∣∣cos−1

(cos(θz) sin(φ)− sin(δ)

sin(θz) cos(φ)

)∣∣∣∣ (2.31)The trough collector is capable of single-axis tracking about the lengthwise axis. This axismay be oriented in any compass direction, though it typically aligns in either the North-Southor the East-West direction. The collector tracks the solar position in such a way that the anglebetween the aperture plane normal and the solar irradiation is minimized. The tracking angleωcol is calculated in Eq.[2.32], where the collector orientation with an azimuth angle (γcol) anda tilt angle (θcol) that is positive when the portion of the field closet to the equator is tilted up[19].

ωcol = tan−1(

cos(θe) sin(γsol− γcol)sin(θe−θcol)+ sin(θcol) cos(θe) (1− cos(γsol− γcol))

)(2.32)

All of the information needed to calculate θ has now been determined. Thus:

θ = cos−1√1− [cos(θe−θcol)− cos(θcol) cos(θe) (1− cos(γsol− γcol))]2 (2.33)

The total radiation incident on the solar field is equal to the available beam-normal irradiation(Ibn) times the total aperture area, but is scaled by the cosine of θ. Thus, this optical derate isreferred to as “cosine loss” and it is the primary variable loss mechanism for the solar field.

Several other solar position losses are modeled, including spillage of reflected radiation off theend of the collector row5, shadowing from one row to another, energy lost before deploying inthe morning and after stowing in the evening while solar energy is still available, and anincident-angle modifier that accounts for all remaining position-dependent losses.

End spillage lossesDuring hours when incoming solar radiation is not directly normal to the collector aperture,some radiation is reflected off the end of the collector that doesn’t reach the receiver. The endloss from each collector can be partially recovered by the adjacent collector, and this energy iscalculated as the “end gain” in Eq. [2.35]. While all collectors incur some end loss, thecollectors that benefit from end gain depend on the solar position as well as the position of the

5A collector loop contains two rows of collectors. See glossary definition for more information.

21

http:Eq.[2.32http:360(2.29

collector within the loop.

For i= 1,Nsca

ηendLoss,i = 1−Lf ,ave,i tan(θ)

Li+ηendGain,i (2.34)

Where:ηendGain,i =

Lf ,ave,i tan(θ)−Lsca,gap,iLi

(2.35)

The ηendGain,i term is equal to zero in the following circumstances:

1. The sun is in the southern sky6 and the collector in question is the southernmost of thecollector row (e.g. i= 1 or i= Nsca)

2. The sun is in the northern sky and the collector in question is the northernmost of thecollector row (e.g. i= �Nsca2 or i= �

Nsca2 +1)

Li is the net total collector length, Lsca,gap,i is the piping distance separating each SCA within asingle row, and L f ,ave,i is the average surface-to-focus path length from the System AdvisorCollectors page. Note that the latter value is not the focal length of the parabola at the vertex,but instead is the total averaged value the light reflecting from the parabolic surface must travelto reach the focus for aperture-normal incidence. Often, only the focal length at the vertex isreadily available, but the averaged value can be calculated using an integral approach, so longas the aperture width (w) is also known.

Beginning with the equation of a parabola y= x2

4a where a is the focal length at the vertex, wecan express the distance traveled from any point (x,y) on the parabola to the focus at (0,a).This arrangement is illustrated in Figure 7. Using the Pythagorean theorem with sides oflength x and y−a, the distance at point (x,y) is:

Lf ,(x,y) =√x2+(y−a)2

=

√x2+

(x2

4a−a)2

The average focal length is then the integral of Lf ,(x,y) with respect to x over the aperture width

6Assuming a field in the northern hemisphere.

22

Figure 7: Focal length geometry for calculating the average focal length.

(−w2 ..+w2 ) and divided by the total aperture width.

Lf ,ave =∫ +w2−w2

√x2+

(x24a −a

)2w

dx

=

√√√√[4a2+ (w2 )2]2a2

·12a2+

(w2)2

12(4a2+(w2)2)

(2.36)

Evaluation of the definite integral results in Eq.[2.36] above, and provides a simple expressionfor average focal length as a function of the focal distance and aperture width.

Row shadowingShadowing between rows generally occurs at extreme solar positions (i.e. dusk or dawn) whenthe shadow cast by a collector closer to the sun obscures a portion of an adjacent collector.Figure 8 shows the geometry associated with row shadowing.

23

http:Eq.[2.36

Figure 8: Two adjacent collector rows may shadow each other if the tracking angle is sufficientlysevere. The shadowing is subject to the collector aperture width, the row spacing (centerline to

centerline), and the tracking angle of the collectors.

The shadowing effect is derived by considering the geometry of the two adjacent collectorrows. Relating the two rows as shown in Figure 8, a right triangle is drawn with a hypotenuseequal to the centerline-to-centerline row spacing Lspacing and short side equal to thenon-shadowed aperture width, wa. To determine the fraction of active collector aperture, wefirst calculate the length of the non-shadowed aperture.

wa = cos(ωcol) Lspacing (2.37)

The shadowing efficiency is equal to the ratio of the non-shadowed aperture to the totalaperture width, w, as shown in Eq. [2.38]. The total shadowing efficiency is limited to therange of values between 0.5 and 1.0. If the shadowing efficiency is less than 0.5, the solar fieldis unlikely to operate successfully, so the total optical efficiency is set to zero and thesimulation progresses to the next time step.

ηshadow =waw

= |cos(ωcol)|Lspacingw

(2.38)

Stow and Deploy AnglesThe user can enforce limits on when the solar field will track the sun. This is given in terms ofthe solar elevation angle, and can be specified for the deploy event and stow event separately.Given a stow angle ωstow and deploy angle ωdeploy that correspond to the solar elevation angle,

24

the time of stow and deploy can be calculated using the same relationship.

tstow/deploy = tnoon+sign(tan(180◦ −ω))

15· cos−1

⎛⎝c1c2+

√c21− c

22+1

c21+1

⎞⎠ (2.39)

where :

c1 =cos(φ)tan(ω)

c2 = −tan(δ)sin(φ)tan(ω)

For the stow or deploy time, ω = ωstow or ω = ωdeploy, respectively. Once the deploy and stowtimes have been calculated, the model calculates the fraction of the time step during with thesystem can operate.

For the deploy hour:

ftrack =tΔt,end− tdeploy

Δt(2.40)

For the stow hour:ftrack =

tΔt,start− tstowΔt

(2.41)

The times at the beginning and end of the time step are given as tΔt,start and tΔt,end ,respectively. The fraction is limited by the model to fall between 0 and 1 duringnon-startup/shutdown time steps.

Incident Angle ModifierThe incident angle modifier ηIAM is a derate factor that accounts for collector apertureforeshortening, glass envelope transmittance, selective surface absorption, and any other lossesthat are a function of solar position. The incident angle modifier factor is calculated using anempirical formula of the form in Eq. [2.42]. The default coefficients for this equation werederived from the field tests of the SEGS LS2 collectors [4]. Equation coefficients a0, a1, anda2 are specified as inputs on the Collectors page.

ηIAM = a0+a1θcosθ

+a2θ2

cosθ(2.42)

(θ in radians)

Constant optical derate factorsThe trough collector model captures optical efficiency with losses that are a function of solarposition and with fixed losses that are applied as constant multipliers. Fixed losses includetracking error, geometry defects, mirror reflectance, mirror soiling, and general error notcaptured by the other items. Because the model multiplies the loss factors together to calculate

25

Table 4: General definitions for each fixed optical loss term

Error description Term Definition

Tracking error ηtrackInability of the collector to perfectly orient along the tracking an-gle; twisting of the collector about the lengthwise axis

Geometry defects ηgeo

Poor alignment of the mirror modules; deviation in the position ofthe receiver tube from the optical focus; warping or discontinuitiesalong the reflective surface

Mirror reflectance ρmSpecular reflectance within a cone angle defined by the collectorand receiver geometry

Mirror soiling ηsoilDirt or soiling on the reflective surface that prevents irradiationfrom reflecting to the receiver

General error ηgen Any effect not captured within the previous categories

an overall loss factor, the value of each individual loss factor is not as important as the value ofthe product of all of the losses. Table 4 describes the physical effect each loss factor isintended to represent.

The total optical efficiency is defined in Eq.[2.43], and we finally calculate the total radiativeenergy incident on the solar field in Eq.[2.44].

ηopt(θ,ωcol) = ηendLoss(θ) ηshadow(ωcol) ηIAM(θ) ηtrack ηgeo ρm ηsoil ηgen (2.43)

q̇inc,s f = Ibn Aap,tot ηopt(θ,ωcol) (2.44)

The incident energy may be adjusted, depending on the stow and deploy times calculatedabove. For situations where the deploy or stow event occurs part way through a time step, thecollector efficiency is reduced by the fraction of time step not in operation. For example, if thesolar field deploys at 7:45am (solar time), the fraction of the time step spent tracking ( ftrack) ismultiplied by the total collector optical efficiency. In this case, ftrack = 1−45/60, or 0.25.

2.3.1 Determining an average efficiency valueSystem Advisor allows the user to assign multiple collector types and geometries within thesame SCA loop (see the Solar Field page). Since each collector variation may have a uniqueoptical efficiency, the reported collector efficiency in the simulation output is equal to theweighted average of all of the collectors used in the loop. System Advisor calculates theweighted optical efficiency by determining the total aperture area of the loop, then multiplyingeach collector’s efficiency value by it’s relative share of the loop aperture. Mathematically, thisis represented in Eq.[2.45] for i collector assemblies in the loop, each with unique area Acol,iand efficiency ηcol,i.

ηcol,ave =Nsca∑i=1

ηcol,i ·Acol,iAloop,tot

(2.45)

26

http:Eq.[2.45http:Eq.[2.44http:Eq.[2.43

Section 2.3 Summary

• The collector field model determines the incoming solar flux by considering weather data,plant location, solar position, and derate values that are both constants and functions ofsolar/collector position.

• Variable losses modeled are end spillage, row shadowing, stow and deploy events, and theincident angle modifier.

• Constant loss fractions account for tracking error, geometry defects, mirror reflectance,mirror soiling, and general error.

2.4 Receivers (HCE’s)The receiver formulation used in the physical trough model uses a 1-dimensional modeldeveloped in [8]. Forristall’s work uses the Engineering Equation Solver (EES) package [15]that is designed to evaluate complex systems of equations using an iterative approach. Thesystem of equations describe the relationship between temperature and heat loss: The surfacetemperature of the absorber tube is a function of the heat absorption, while convective andradiative losses are strong functions of surface temperature. Convective loss is directlyproportional to the difference between receiver surface and ambient temperatures, andradiation loss is proportional to temperature difference to the fourth power. The performanceof the receiver can’t be accurately modeled using simple explicit relationships. Instead, SystemAdvisor uses implicit equations, solving iteratively with successive substitution until thesolution converges.

The receiver is modeled as a 1-dimensional energy flow. Only the temperature gradient in theradial direction is assumed to be significant - axial and circumferential heat transfer areneglected. Figure 9 presents a diagram of one quarter of the receiver in cross-section. Eachtemperature T1−5 is calculated by the model using an energy balance andtemperature-dependent loss coefficients. The receiver geometry is specified by the user withradii R1−4.

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Figure 9: A heat balance for the modeled receiver. Heat transfer in the radial direction (left toright) is considered, while circumferential and axial transfer is not.

Concentrated irradiative flux from the collector passes through the transparent glass envelope(R3−4), and a portion of that flux is absorbed by the glass. The absorbed flux is determinedfrom the absorption fraction specified by the user as Envelope Absorptance (αenv) on theReceivers page and influences the calculated temperature of the glass. The flux that passesthrough the glass envelope reaches the absorber tube at R2. Note that the fraction of energypassing through the glass envelope is specified by the Envelope Transmittance value on theReceivers page, and need not equal the compliment of the absorptance value. This is becauseabsorptance by the glass is only one of several possible loss mechanisms. Others includereflective loss and light refraction where incoming rays are bent away from the absorber.

During operation, the heated surface at R2 drives thermal energy through the absorber wall(R1−2) and into the cooler HTF. Thermal losses from the absorber surface occur via convectionand radiation exchange with the glass envelope. The glass envelope is in turn exposed toambient air. Figure 10 shows the heat transfer network, conceptualized as a set of thermalresistances in series and parallel. This is analogous to an electrical resistance network wherethermal energy represents current, thermal resistance represents electrical resistance, andtemperature drop is equivalent to voltage drop. Incidentally, the same resistance formulaeapply to thermal and electrical networks.

2.4.1 Modeling approachThe nodal nature of the collector loop was discussed in Section 2.1 (see Figure 2 on page 8).To summarize, each node corresponds to an assembly of individual receiver elements andcollector modules. As HTF passes through the loop, it gradually warms until it reaches thedesign-point field outlet temperature at the end of the last SCA. The gradual warming of HTFover the length of the loop corresponds to a trend of decreasing thermal efficiency, since

28

Figure 10: The thermal resistance network for the “intact” receiver model shown in Figure 9.Energy is absorbed at T3 and T4−5.

Figure 11: The thermal resistance network for the “broken glass” receiver model. Energy isabsorbed on the absorber tube surface at T3 and heat is exchanged directly with the sky and

ambient temperatures.

receiver performance is inversely related to receiver temperature. Variability in receiverperformance within a loop can be significant, so the receiver model is applied individually foreach node in the loop. This system is solved iteratively to determine the mass flow rate that isrequired to meet the design outlet temperature, as discussed in Section 2.2.

The receiver model uses information about the HTF temperature, receiver geometry, ambientconditions, and incoming solar flux to determine the performance of the receiver.Conceptually, the solar field can be dissected into four different models:

1. the collector model

2. the receiver model

3. the piping model, and

4. the HTF model.

This distinction is particularly noteworthy for the receiver and HTF models. The HTF modelcalculates the HTF temperature throughout the loop based on absorbed energy and mass flowrate. The receiver model calculates the thermal performance of the receiver given an HTFtemperature and other information. Thus, from the perspective of the receiver model, the HTF

29

temperature (T1) is an input value even though T1 is closely tied to the results calculated by thereceiver model. Other specified values for the model are summarized in Table 5.

Table 5: Inputs to the receiver model

Item DescriptionT1 HTF inlet temperatureṁht f HTF mass flow rateTamb Ambient temperatureTsky Effective sky temperaturevwind Wind velocity at the receiver surfacepamb Ambient pressureq̇inc,i Incident radiation at node iAcs Cross-sectional area of the absorber tubeD2 Absorber tube internal diameterD3 Absorber tube external diameterD4 Glass envelope internal diameterD5 Glass envelope external diameterDp Internal flow plug diameterε3 Absorber surface emittanceε4 Glass surface emittanceαabs Absorber surface absorptanceαenv Glass envelope absorptanceηcol,i Collector optical efficiency at node iτenv Glass envelope transmittancePa Annulus pressure- Annulus gas type- HTF type- Absorber material

For any solver using successive substitution, initial guess values must also be provided. Theguess values for the receiver model are initially calculated based on the HTF temperatureprovided to the model, and depend on the condition of the receiver envelope. Temperatureguesses for the absorber tube and glass envelope must be provided. Eq.[2.46] shows the initialsettings for the temperatures for intact receivers, and Eq.[2.47] shows the settings for receiverswith broken glass.

T2 = T1+2◦CT3 = T2+5◦C (2.46)T4 = T3−0.8 · (T3−Tamb)T5 = T4−2◦C

T2 = T1+2◦CT3 = T2+5◦C (2.47)T5 = T4 = Tamb

30

http:Eq.[2.47http:Eq.[2.46

Once guess values have been calculated, subsequent calls to the subroutine use the convergedvalues from the previous call as the new guess values. However, several conditions may triggerrecalculation of the guess values using Eq.’s[2.46] and [2.47]:

• The difference between the last T1 and the current T1 is greater than 50◦C

• The minimum value in the group T1−5 is less than the current Tsky value

• Any temperature from the last call returned as invalid (Not a Number error)

2.4.2 Model formulationThe first step in determining receiver heat loss is to calculate the thermal resistance betweenthe outer absorber tube and the inner glass envelope surfaces. Both convection and radiationcontribute to the total heat transfer, though convection between the two surfaces is very smallfor intact receivers. Convection becomes significant in cases where the vacuum is lost due tobroken glass or where hydrogen from the HTF has diffused through the absorber tube wall intothe annulus.

Convection from the absorberConvection may occur either between the absorber and the inner glass surface or directly to theambient air in the case that the envelope is broken. The convection subroutine handles bothsituations. First, for intact receivers, the annulus gas properties are evaluated at the averagetemperature T34. Convection from R3 to R4 can be generally expressed in terms of thermalresistance R̂34,conv as7:

q̇34,conv =T3−T4R̂34,conv

(2.48)

Where:R̂34,conv =

1γ34,conv π D3

The receiver model calculates natural internal convection using the modified Raithby andHollands correlation [2] (for more information on the convective algorithms, see [8] pages11-14). The calculation for annular natural convection begins with determining the Rayleighnumber at diameter D3 using Eq.[2.49].

RaD3 =g β34 |T3−T4|D33

α34 ν34(2.49)

The volumetric expansion coefficient β34, the thermal diffusivity constant α34, and thekinematic viscosity ν34 of the annular gas are all evaluated at the averaged temperature T34.

7The subscript “34” (or other two-number subscripts in this section) denote properties, temperatures,or other quantities that describe the intermediate step between surface 3 and surface 4 in the resistancemodel. In practical terms, “34” can be thought of as the subscript that denotes the thermal interaction ofsurface 3 and surface 4. This model describes the continuous substance between surfaces 3 and 4 usingthermal properties that are evaluated at the average of T3 and T4 (T34).

31

http:Eq.[2.49http:Eq.�s[2.46

Using Prandlt number Pr34 = ν34/α34, we calculate the heat transfer due to natural convectionin the annulus and the associated heat transfer coefficient.

q̇34,conv = 2.425 k34T3−T4(

1+ D3D40.6)1.25

(Pr34RaD30.861+Pr34

)0.25(2.50)

γ34,conv =q̇34,conv

π D3 (T3−T4)(2.51)

For very low annular pressures, the molecular density drops below the physical limit forestablishing convective currents; instead, free molecular heat transfer relationships moreappropriately describe convective heat loss. The receiver model handles this by using thelargest convective loss predicted by either annular natural convection or free molecular heattransfer. Eq.[2.52] shows the steps for calculating free molecular heat transfer.

Λ = C1×10−20 ·T34Pa ·ζ2

Γ =cp,34cv,34

b =9Γ−52Γ+2

γ34,conv =k34

D32 log

(D4D3

)+ b Λ100

(D3D4 +1

) (2.52)q̇34,conv = π D3 γ34,conv (T3−T4)

In the calculation for Λ,C1 is a constant 2.331×10−20mmHg·cm3

K , ζ is the free-molecularcollision distance shown in Table 6 [8], and Pa is the annulus pressure in torr.

Table 6: Values of the mean free path between collisions of a molecule for free molecularconvection

Annulus Gas ζ [cm]Air 3.53×10−8

Hydrogen 2.4×10−8Argon 3.8×10−8

The annular convection calculations assume that the receiver’s glass envelope is intact.However, the glass envelope sometimes breaks due to impact or excessive thermal cycling.Thermal loss from a broken-glass receiver is significantly higher than for an intact receiver,and the loss must be modeled differently. System Advisor provides specialized calculations forbroken-glass receivers and further divides the heat transfer relationships applied based onambient wind speed.

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http:Eq.[2.52http:Pr34RaD30.25

If the ambient wind speed is very low (less than 0.1 m/s), then the Nusselt number is calculatedusing the Churchill & Chu correlation for a long isothermal horizontal cylinder [13], where thefluid properties are determined at the averaged temperature T36.

Nu=

⎡⎢⎢⎢⎣ 0.60+0.387 ·Ra

0.1667D3(

1+(0.559Pr36

)0.5625)0.2963⎤⎥⎥⎥⎦2

(2.53)

The convection coefficient calculated in Eq.[2.54] is then used to determine the totalconvective heat transfer.

γ34,conv = Nuk36D3

(2.54)

q̇34,conv = γ34,conv π D3 (T3−T6)

If the ambient wind speed is above 0.1 m/s, thermal properties are required for both the air incontact with the absorber surface at T3 and the ambient air at T6. In this case, the Nusseltnumber is calculated using Zhukauskas’ correlation for external forced convection [13].

Nu=C RemD3 Prn6

(Pr6Pr3

)0.25(2.55)

Where:ReD3 =

v6 D3ν6

The coefficients m, n, and C are selected according to the Prandtl number and the Reynoldsnumber as shown in Table 7. For Pr ≤ 10, n= .37, otherwise n= .36.

Table 7: Selection of coefficients C and m for Zhukauskas’ correlation based on the Reynoldsnumber at D3.

Reynolds Number Range C m0 ≤ ReD3 < 40 0.75 0.440 ≤ ReD3 < 1000 0.51 0.5

1000 ≤ ReD3 < 2×105 0.26 0.62×105 ≤ ReD3 < 106 0.076 0.7

To summarize the absorber convection calculations:

1. Convection loss is determined based on the condition of the receiver (whether the glassenvelope is intact or broken).

33

http:Eq.[2.54

2. If the glass is intact, the natural convection coefficient and the molecular diffusionconvection coefficient are both calculated and compared, with the larger of the twoselected for use in the loss equation.

3. If the envelope is not intact, direct convection to ambient is calculated based on whetherthe wind speed is below or above 0.1 m/s.

No matter the method used to calculate the convective loss coefficient from the absorber, thethermal resistance due to convection is expressed as follows.

R̂34,conv =1

γ34,conv π D3(2.56)

Radiation from the absorberRadiation loss from the absorber tube to the surroundings is the most significant contributor toheat loss for intact collectors. Two alternate equations are used for calculating radiative lossdepending on whethe